The inclusion of the strain is believed to greatly increase the bandwidth at the magic angle [1]}, [2]}, [3]}. Having seen that the relaxed atomic configuration of the twisted bilayer graphene obtained in Sec.  expands the AB/BA stacked regions and shrinks the AA stacked regions relative to just a rigid twist,
thus intrinsically inducing strain, it is interesting to ask whether the bandwidth undergoes the increase as well.
Motivated by this question, the goal of this section is to generalize the chiral limit introduced and analyzed in Ref. [4]} for the BM model including the relaxation induced pseudo-magnetic vector potential \(\mathcal {A}\) .
While the relaxation induced \(\mathcal {A}\)  indeed increases the vanishing bandwidth at the magic angle found in Ref. [4]} without \(\mathcal {A}\) , we demonstrate below that decreasing the twist angle can compensate the effect of \(\mathcal {A}\)  on the bandwidth, resulting in exactly flat bands at the CNP at a new (smaller) magic angle. Throughout our analysis we pay particular attention to the importance of \(C_3\)  symmetry (preserved by \(\mathcal {A}\) ) in making the compensation possible, noting that extrinsically induced strain generally breaks \(C_3\) .
